I am curious about the submodules of a module with a given property. Let $M$ be an $R$-module.
If $M$ is a finitely generated are the submodules of $M$ finitely generated?
If $R=\mathbb Z$, $M$ is a abelian group. The subgroup of $M$ is finitely generated by Proving that a subgroup of a finitely generated abelian group is finitely generated
If $R$ is Noetherian, we also have positive answer.
I'd like to know is there any generalization that contains the situation $R=\mathbb Z$ or Noetherian.
Any advice is helpful. Thank you.